Infinite impulse response (IIR) digital filter structures have been studied for more than thirty years. IIR filters generally fall into two types, cascade and parallel. The parallel IIR filter theoretically provides less noise, i.e., provides a higher signal-to-noise ratio. However, for two major reasons most IIR filters are nonetheless implemented using the cascade form. First, the parallel form is considered to be computationally expensive. See, for example, L. Jackson, Digital Filters and Signal Processing, Kluwer Academic Publishers, 1986, at page 202. The cascade IIR filter requires 25-50 percent fewer multiply operations than required for the parallel IIR filter, thus saving processing hardware and instruction cycles. For these practical reasons, the cascade form is often preferred.
Second, known parallel IIR filters suffer from reduced performance due to quantization errors. These quantization errors reduce signal-to-noise ratio (SNR) and can arise from two sources. First, commercially-available digital signal processors (DSPs) can create quantization errors when implementing the biquad filters of the IIR filter. The microelectronics revolution has provided DSPs with sophisticated data addressing modes and high arithmetic precision. Nonetheless, commercially available DSPs do not provide the precision required to avoid creating quantization errors. While these DSPs perform the repetitive arithmetic operations required for IIR filters in a high-precision arithmetic logic unit (ALU), sometimes the precision is lost. For example, the high operand precision is lost when an operand must be stored in memory to implement a time delay. This lost precision results in quantization error.
Second, quantization errors occur when the IIR filter polynomial is decomposed to determine the parallel form coefficients. The desired transfer function of the filter is easily expressed in cascade form, that is, as a product of some number of biquad terms. Determination of cascade-form coefficients is easy because only second-order decomposition of each biquad term is involved. In theory, the parallel form of the transfer function can be obtained directly from the cascade form. However, in practice it is very difficult to generate parallel-form filter coefficients directly from the cascade form transfer function. Conversion from the given cascade filter coefficients to the parallel filter coefficients involves a large number of multiplications and divisions. As the number of computations increases, quantization errors accumulate. The coefficient inaccuracies due to the accumulated quantization errors may also cause stability problems. In the parallel IIR filter, the placement of the filter's zeros depends on cancellation of terms in a summation operation, and thus is more sensitive to coefficient quantization. Inaccurate coefficient quantization may cause the zeros to move off of the unit circuit and ultimately cause the parallel-form IIR filter to be unstable.
Thus, both conventional methods to decompose the polynomial and sum the decomposed terms, and known biquad filter structures, require very high precision processing. When very high precision processing is not practical, quantization errors result, lowering performance below theoretical potential. What is needed, then, is a new parallel IIR filter which solve the aforementioned problems and may be implemented using commercially available DSPs with minimum cost.